Relative positions and tangency

For radii R,r>0 and centre distance d, joining the centres to any common point makes a triangle. Its sides require |R-r| <= d <= R+r. Equality flattens the triangle and gives contact; strict inequalities give two intersection points. Method: find both centres and positive radii, calculate d, then compare with R+r and |R-r|. Check d=0 separately: equal radii mean coincident circles, unequal radii mean no common point. Example: R=5, r=2. At d=7 the circles touch externally; at d=3 they touch internally. At d=1 the smaller circle is strictly inside the larger and they do not meet. Exam check: internal tangency needs d=|R-r|>0. Equal circles with the same centre coincide; they are not internally tangent. For contact coordinates, use the line of centres and verify the point in both equations.